Signals and Systems - Electrical Engineering

Problems 351

Table 5.1Basic Properties of the Fourier Transform

Time Domain Frequency Domain

Signals and constants x(t),y(t),z(t),α,β X(),Y(),Z()

Linearity αx(t)+βy(t) αX()+βY()

Expansion/contraction in time x(αt),α6= (^0) |^1 α|X
(
α
)
Reflection x(−t) X(−)
Parseval’s energy relation Ex=
∫∞
−∞|x(t)|
(^2) dt Ex= 1
2 π
∫∞
−∞|X()|
(^2) d
Duality X(t) 2 πx(−)
Time differentiation d
nx(t)
dtn,n≥1,integer (j)
nX()
Frequency differentiation −jtx(t) dXd()
Integration
∫t
−∞x(t
′)dt′ X()
j +πX(^0 )δ()
Time shifting x(t−α) e−jαX()
Frequency shifting ej^0 tx(t) X(− 0 )
Modulation x(t)cos(ct) 0.5[X(−c)+X(+c)]
Periodic signals x(t)=
∑
kXke
jk 0 t X()=∑
k^2 πXkδ(−k^0 )
Symmetry x(t)real |X()|=|X(−)|
∠X()=−∠X(−)
Convolution in time z(t)=x∗y Z()=X()Y()
Windowing/multiplication x(t)y(t) 21 πX∗Y
Cosine transform x(t)even X()=
∫∞
−∞x(t)cos(t)dt,real
Sine transform x(t)odd X()=−j
∫∞
−∞x(t)sin(t)dt,imaginary
Table 5.2Fourier Transform Pairs
Function of Time Function of
1 δ(t) 1
2 δ(t−τ) e−jτ
3 u(t) j^1 +πδ()
4 u(−t) −j^1 +πδ()
5 sgn(t)=2[u(t)−0.5] j^2 
6 A, −∞<t<∞ 2 πAδ()
7 Ae−atu(t),a> (^0) jA+a
8 Ate−atu(t),a> (^0) (jA+a) 2
9 e−a|t|,a> (^0) a (^22) +a 2
10 cos( 0 t), −∞<t<∞ π[δ(− 0 )+δ(+ 0 )]
11 sin( 0 t), −∞<t<∞ −jπ[δ(− 0 )−δ(+ 0 )]
12 A[u(t+τ)−u(t−τ)],τ > 0 2 Aτsinτ(τ)
13 sin(πt^0 t) u(+ 0 )−u(− 0 )
14 x(t)cos( 0 t) 0.5[X(− 0 )+X(+ 0 )]